Is there an advantage to laminating over not?
Oh yeah! (I mean, there can be, if that's your thing.) When you do it right, laminated cracker style pizzas are unbelievable. Even when you don't do it right, they can still be unbelievable. The texture of a laminated cracker style pizza done right is unlike any other kind of pizza. Crispy (or crunchy) on the bottom, then progressively closer to chewy/bready with each layer above the bottom layer.
And there are so many different ways to do it. Fazzari does it with very low hydration. I think he has said he uses maybe as low as 33% hydration + 4% fat at work. I assume that's very crunchy and unlike anything I've ever had. Then you have Shakey's, which is probably about the same, but with a little higher hydration (38-40% +2-4% fat). So it's not as crunchy as fazzari's but still crunchy. Then you have my Tommy's clone, which is 43% hydration + 5% fat. This pizza is crispy/flaky at the bottom and can be chewy at the top. And finally you have Round Table, which I guess is something like 48% hydration + 2-4% fat. Crispy at the bottom, then chewy, then probably a little bready at the top. Round Table may not be a true cracker style, but I don't think anyone has a problem with it being considered cracker style, I guess because laminated crust is kind of a style of its own.
Your dough, I would say, is in Round Table territory. With softer dough like yours, it's probably best to use some bench flour between the layers. Otherwise, the layers will probably merge right back together when you roll the folded dough. With fazzari's or Shakey's style, however, there's no way you want to use any bench flour. With Tommy's dough, I've gotten to where I dust the layers with a very minimal amount of bench flour. You could even call it none.
Aside from what I've already said, there are many more slight changes you can make to end up with different but equally awesome cracker style pizzas. Like adding more fat. Beyond 5% it starts getting flakier, from my experience. There are different ways to laminate and ferment, too. You can use AP or high gluten flour, and they create equally great but different results.
Different thicknesses can create vastly different pizzas with the same dough, using the same number of layers. Flour or no flour between the layers?
Even though I never considered myself a particularly huge fan of cracker style vs. other styles, that changed shortly after member briterian asked if anyone knew how to make Tommy's pizza. I didn't have a clue back then, but I tried it anyway. Once that happened, I was changed forever.
Here's a bit of info that took me a long time to figure out: Laminated cracker pizzas would rather never touch a pan (except when you use a pan as a template to trim the dough). I have a lot of different styles and sizes of pans, but the only ones I ever use anymore are the ones that are necessary to hold the shape of a skin. Basically deep dish pans for deep dish and stuffed, because you can't make either of those styles of pizza without a pan. Most other styles can be (and should be) baked directly on stone without pans. My opinion, of course.
Also, I think parbaking pizza skins is almost always a waste of time, for many reasons. I'll save my explanation for another post, though.